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Ind. Eng. Chem. Process Des. Dev. 1980, 19, 505-507
Literature Cited Benham, A. L., Katz, D. L., AIChE J., 3, 33 (1957). Chao, K. C., Lin, H. M., Simnick, J. J., Sebastain, H. M., references for H, hydrocarbon and CH, hydrocarbon systems were ctted in Annual Report, EPRI Project RP 367-2, 19'78; results for COS-containingmixtures are unpublished. Chueh, P. L., Prausnitz, J. M., Ind. Eng. Chem. Fundam., 6, 492 (1967). Connolly, J. F., J. Chem. Phys., 36, 2897 (1962). Connolly, J. F., Kandalic, G. A., API 28th Midyear Meeting, Philadelphia, PA, May 13, 1963 (Paper 14-63), DeVaney, W. E., Kao, P. L., Borryman, J. M., Progress Report to GPA, 1977. Graboski, M. S., Daubert, T. E., Ind. Eng. Chem. Process Des. Dev., 17,443 (1978a). Graboski, M. S., Daubert, T. E., Ind. Eng. Chem. Process Des. D e v . , I ? , 448 (1978b). Graboski, M. S., Daubert, T. E., Ind. Eng. Chern. Process D e s . Dev., 18, 300 (1979). Gray, R. D., "Correlation of H,/Hydrocarbon VLE Using Redlich-Kwong Variants", presented at 70th Annual AIChE Meeting, New York, 1977. Kiink, A. E., Cheh, H. Y., Amick, E. H., AIChE J., 21, 1142 (1975). McCarty, R. D., "Hydrogen Technological Survey-Thermophysical Properties", NASA-SP-3089 (1975).
+
+
Oiiphant, J. L., Lin, H. M., Chao, K. C., Fluid Phase Equilib., 3 , 35 (1979). Peter, S., Reinhartz, K., Z. Phys. Chem., 24, (1960). Reed, T. M., Gubbins, K. E., "Applied Statistical Mechanics", Chapters 7 and 9, McGraw-Hiii, New York, 1973. Reid, R. C., Prausnltz, J. M., Sherwood, T. K., "The Properties of Gases and Liquids", 3rd ed, Appendix A, pp 629-665, McGraw-Hill, New York, 1977. Sebastian, H. M., Lin, H. M., Chao, K. C., submitted to J. Chem. €no. Data for publication (1980). Simnlck, J. J., Sebastian, H. M., Lin, H. M., Chao, K. C., J. Chem. Eng. Data, in press (1980). Soave, G., Chem. Eng. Sci., 27, 1197 (1972). Spano, J. O., Heck, C. K., b r i c k , P. L., J. Chem. Eng. Data, 13, 168 (1968). Trust, D. 9.. Kurata, F., AIChE J., 17, 86 (1971). Yorizane, M., Yoshimura, S., Masouka, H., Kagaku KOgakU, 34, 953 (1970).
School of Chemical Engineering Purdue University West Lafayette, Indiana 47907
Ho-mu Lin
Received for review August 10, 1979 Accepted M a r c h 24, 1980
Distillation Calculation for Cases of Specified Heat Duty by Modification of Existing Computer Programs Some algorithms applied to solve problems of distillation in which heat duty of the reboiler or the condenser is specified, but the reflux ratio is not, are discussed. A useful and flexible algorithm is presented, in which an existing computer program for an operating column analysis is employed without change, and a useful correction loop for the reflux ratio is added externally. This loop is obtained from the relationship between the reflux ratio and an overall energy balance. This algorithm is readily applied because It is independent of the calculation technique used with an existing computer program. Numerical examples indicate the effectiveness and reliability of this algorithm.
Introduction In an operating distillation column analysis, the reflux ratio is usually specified. However, we must sometimes solve problems of multicomponent distillation in which the heat duty of the reboiler or the condenser is specified instead. Some algorithms, which are applicable to problems like this, are discussed. We then propose a simple algorithm in which an existing computer program for an operating column analysis is employed without any change. In this algorithm, the correction loop for the reflux ratio, which is obtained from the relationship between the reflux ratio and the energy balance of a distillation column, is added to the outside of an existing computer program for an operating column analysis. This algorithm is independent of the calculation technique of the existing computer program, e.g. the Newton-Raphson method (Naphtali and Sandholm, 1971; Kubichek et al., 1976; Kawase et al., 1978), the tridiagonal matrix method (Wang and Henke, 1966), and so on. Therefore, this algorithm is flexible and readily applied. Its effectiveness is discussed by numerical examples for the lower column of an air separation plant. Algorithm and Calculation Procedure For solving distillation problems in which the heat duty of the reboiler or the condenser is specified but the reflux ratio is not, some algorithms may be available. First, consider algorithms in which it is necessary to revise an existing computer program. In the case of the NewtonRaphson method, we must replace the dependent variable Q, or QR with L1(or R ) (Kubichek et al., 1976). When the dependent variable Q, is replaced by L1,this problem is easily solved by using the Thomas algorithm without revision because of the block tridiagonal form of its Jacobian matrix. However, we must change the computer program 0196-4305/80/1119-0505$01.00/0
according to the replacement of the dependent variable. When QR is specified and replaced as a dependent variable by L1,an off-band element occurs in the Jacobian matrix. Therefore, any existing computer program must be drastically changed because it is impossible to apply the Thomas algorithm directly. Instead, a modified Thomas algorithm to solve this problem must be employed (Kubichek et al., 1976; Waggoner and Loud, 1977; Hofeling and Seader, 1978). In the case of the tridiagonal matrix method (Wang and Henke, 19661, the procedure for evaluating the vapor flow rate Vi and the liquid flow rate Lj by the energy balance equation and the overall material balance equation must be changed. A schematic representation of the j t h plate of the distillation column is shown in Figure 1. When Q, is specified but L1(or R ) is not, the computational procedure for obtaining new values of V; and Li is (step 1) Ll = -iQc + V1Wl - H2)l/(hl - H2)
(1)
(step 2) v, = v1 + L1 (2) (step 3) Vj+l = [(Vj+ Wj)(Hj- hj) - Lj-l(h;-l- hj) - Fj(H., hi) + Qjl/(Hj+~ - hj) (3) 0'=2,N-l) (4)
0'=2,N-l) where F1= W1 = 0 and D = Vl + U1. Values for Liand Vj are successively obtained from plate 1 to plate N by using eq 1 to 4. In a usual operating column analysis, where L1(or R) is specified, eq 1 is not necessary. 0 1980 American
Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980
,
,
U
JLj? jti
th
Figure 1. Conditions of j t h plate.
)-(
N-I@
a Set initial values
l~sxlmerettux ratio
VN
I
Figure 3. Distillation column. D ;02509moles
distillation calculation
( a n existi4 p w r a m
m
Condenser
Imp A )
e correct reflux ratio (Eq 19)
Figure 2. Calculation procedure flow chart.
When QR is specified instead of L1 (or R ) , the computational procedure for obtaining new values of V j and Lj is (step 1) V N = (QR - LN(hN - hN-i)]/(" - hwi) (5) LN-1 = VN + LN (6) (step 2) When these values are available, a recursive operation may be used to obtain succeeding values of L j and V,.
i-1
Figure 4. Statements of the lower column of an air separation plant.
the distillation calculation. This procedure is continued until the convergence criterion is satisfied. We propose a very convenient and useful method for correcting the reflux ratio in this algorithm. Figure 3 shows parts of the top and bottom of a conventional distillation column. The plates are numbered from the top of the column to the bottom with the condenser as the first plate and the reboiler as the Nth plate. The total distillate rate, D , and Vl and Ul, if necessary, are specified. The energy balance around the condenser is Q, = V2H2 - V1H1 - Llhl - Ulhl (9) where L1 and V 2are given by L1 = R(V1 + Vi) = RD V 2 = D L1 Substituting eq 10 and 11 into eq 9 we obtain Q, = (1 R)DHz - ViH1- RDhl- Ulhl The derivative of the heat duty of the condenser respect to the reflux ratio is represented as
+
In this case, values of Lj and V j are evaluated from the bottom to the top of the column. Consequently, we must revise an existing computer program for an operating column analysis. We investigated this algorithm by numerical examples of a distillation column for an organic acid separation process. Secondly, consider an algorithm in which it is not necessary to change an existing computer program. The flow chart for this algorithm is shown in Figure 2. We assume an initial value of the reflux ratio and the computer program for an operating column analysis as if the reflux ratio is specified but the heat duty of the reboiler or the condenser is not. After converging this calculation, the heat duty obtained from the calculation, QCL,is compared with the specified one, QC. If the calculated value agrees with the specified one within the tolerance, t, we have the final solution. If not, we must correct the reflux ratio and repeat
+
(10) (11) (12) with
Assuming that the change of enthalpy caused by a change in the reflux ratio is negligible, we obtain the following approximate relationship. dQ,/dR DH2 - Dhl (14) The energy balance of the entire column is AQF + QR = 8, + QL
(15)
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 3, 1980
Table 11. Typical Convergence Behavior of Numerical Numerical Examples
Table I. Iteration Number of Loop A and B vs. Initial Value of Reflux Ratio for Numerical Example Aa
B
Aa
B
1 2 3 4 5 6 7 8 9 10
58 5 01 31 45b 44 45 45 45 46 47
2 1 1 2 2 2 2 2 2 2
74b 119 64 45 61 61 62 62 63 63
3 5 3 2 3 3 3 3 3 3
18
0
18
0
Sum of Loop A.
iteration no. (loop B)
E = l
-E
2.402c a
= 30
initial reflux ratio
See Table 11.
507
0 1 2
R QCL, J Case 1. IQG- Q c ~
